Establish the general self-affine dimension formula in higher dimensions

Establish equality between Hausdorff dimension and affinity dimension for general self-affine systems in dimensions d≥4, beyond the currently available planar and restricted higher-dimensional results.

Background

The paper explains that planar self-affine dimension results follow from projection theorems for Furstenberg measures on the projective line. It also notes that higher-dimensional extensions face a new obstruction: Furstenberg measures may arise from non-conformal dynamics and may concentrate locally near hyperplanes, causing the spread-outness hypothesis to fail.

The unresolved problem is the general higher-dimensional analogue of the dimension formula, namely proving that the Hausdorff dimension of a self-affine set or measure is governed by its affinity dimension under appropriate general hypotheses in dimensions at least four.

References

The general problem remains open for $d\ge 4$. Our method has an obstacle in higher dimensions: the Furstenberg measures are invariant under non-conformal dynamics and can concentrate locally near hyperplanes, so spread-outness can fail.

— Spread-out Furstenberg sets II: general configurations and fractal restricted projections  (2610.13682 - Cohen et al., 9 Oct 2026) in Section “Dimension of planar self-affine sets and measures,” final paragraph